Good thought experiment!

I think part of the problem is that our identities are partly socially decided. In my theory of use I use the thought experiment of Monday-Tom and Tuesday-Tom, and explain why, although they undergo changes, they are seen as the same Tom, but also how, if the change is significant (complete memory whipeout, for instance), they are no longer seen as the same Tom.

In a similar fashion, let's say I got a glass cup that I use to drink water and it's now fractured and leaking. So I no longer use it to drink water, cos it can't hold it. It's essentially no longer a glass cup. More like just glass. Similarly, a "woman" is something that can be used by males in society in a certain way. Among those uses is childbearing. Among those uses is "the way my family reacts to her", etc. So a woman similarly might call a man "not a real man" because one of his uses is, oh I don't know, "risking his life to save her" or "financial provider" etc., so any time she can't use him the way she wants or expects to, she describes her feeling as "you're not a real man" etc.

So if we define objects according to how we use them, we will remove their identity when they can't be used in the "proper" way.

But if technology comes along and makes brains flexibly useable (can put them in different bodies willy-nilly), then I expect our concept of identity will broaden to include all those new uses.